Online Trigonometry Calculator: 100% Free

Key Features:

1. Trigonometric Functions Calculator:
   • Calculates sine, cosine, tangent and their inverses
   • Handles degree and radian conversions
   • Real-time calculations
2. Right Triangle Solver:
   • Solves for missing sides/angles using Pythagorean theorem
   • Accepts any two known values
   • Calculates remaining side and angles
3. User Interface:
   • Responsive grid layout
   • Clean, modern design with card-based interface
   • Input validation
   • Clear result display
4. Visual Features:
   • Unit conversion between degrees and radians
   • Error handling for invalid inputs
   • Interactive input fields

How to Use:

1. Trigonometric Functions:
   • Select function from dropdown
   • Enter angle in degrees
   • Click “Calculate”
2. Right Triangle Solver:
   • Enter any two known sides
   • Leave unknown side as 0
   • Click “Solve Triangle”
3. Unit Conversion:
   • Enter value in degrees or radians
   • Click conversion buttons

Technical Specifications:

• Uses JavaScript’s Math library for calculations
• Responsive design with CSS Grid and Flexbox
• Input validation and error handling
• Clean separation of HTML, CSS, and JavaScript
• Cross-browser compatible

Enhancements:

1. Add angle visualization using CSS/SVG
2. Implement hyperbolic functions
3. Add graph plotting capabilities
4. Include trigonometric identities calculator
5. Add complex number support

This calculator provides basic to intermediate trigonometric calculations while maintaining a user-friendly interface suitable for students and professionals.

Comprehensive Guide to Trigonometry

Trigonometry is a branch of mathematics that studies relationships between angles and sides in triangles. Below is a detailed breakdown of its concepts, formulas, and applications:

What is Trigonometry?

Trigonometry (from Greek trigonon = triangle, metron = measure) focuses on:

• Calculating unknown sides/angles in triangles.

• Analyzing periodic phenomena like sound waves and tides.

• Solving real-world problems in engineering, physics, and astronomy.

Trigonometric Functions

Six primary functions for right-angled triangles:

Function	Definition	Reciprocal	Formula
Sine (sin)	Opposite / Hypotenuse	Cosecant (csc)	$\sin \theta =\frac{O}{H}$
Cosine (cos)	Adjacent / Hypotenuse	Secant (sec)	$\cos \theta =\frac{A}{H}$
Tangent (tan)	Opposite / Adjacent	Cotangent (cot)	$\tan \theta =\frac{O}{A}$

Mnemonic: SOH-CAH-TOA (Sine = Opposite/Hypotenuse, etc.).

Unit Circle & Angles

The unit circle (radius = 1) simplifies trigonometric calculations:

• Coordinates: For angle $\theta$, $(\cos \theta ,\sin \theta )$.

• Quadrant Rules: Sign of functions depends on the quadrant.

• Special Angles:

  • $0^{\circ },30^{\circ },45^{\circ },60^{\circ },90^{\circ }$.

  • Exact values (e.g., $\sin 30^{\circ }=\frac{1}{2}$).

Trigonometric Identities

Essential Identities:

1. Pythagorean:
   ${\sin }^2\theta +{\cos }^2\theta =1$.

2. Sum/Difference:
   $\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$.

3. Double Angle:
   $\sin 2\theta =2\sin \theta \cos \theta$.

4. Half Angle:
   $\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}}$.

Trigonometric Equations

Solve equations like $\sin x=0.5$:

• General Solutions:
  $x=\theta +2\pi n$ or $x=\pi -\theta +2\pi n$ (for sine).

• Principal Values:
  Use inverse functions (${\sin }^{-1},{\cos }^{-1},{\tan }^{-1}$).

Graphs of Trigonometric Functions

Function	Graph Features
Sine	Periodic (period $2\pi$), range $[-1,1]$
Cosine	Similar to sine, shifted $\frac{\pi }{2}$.
Tangent	Vertical asymptotes at $\frac{\pi }{2}+n\pi$.

Applications of Trigonometry

• Physics: Calculating forces, projectile motion.

• Engineering: Structural design, robotics.

• Astronomy: Measuring distances between stars.

• Navigation: GPS systems, ship/aircraft routing.

• Music: Sound wave analysis.

Laws of Sines and Cosines

For non-right triangles:

1. Law of Sines:
   $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

2. Law of Cosines:
   $c^2=a^2+b^2-2ab\cos C$.

Inverse Trigonometric Functions

Used to find angles from ratios:

• ${\sin }^{-1}x$, ${\cos }^{-1}x$, ${\tan }^{-1}x$.

• Domains:

  • ${\sin }^{-1}x$: $[-1,1]\to [-\frac{\pi }{2},\frac{\pi }{2}]$.

  • ${\cos }^{-1}x$: $[-1,1]\to [0,\pi ]$.

Real-World Examples

1. Height of a Building:
   $\text{Height}=\text{Distance}\times \tan (\text{Angle of Elevation})$.

2. AC Circuits:
   Voltage $V(t)=V_0\sin (\omega t+\varphi )$.

Common Mistakes to Avoid

• Confusing degrees and radians.

• Misapplying identities (e.g., $\sin (A+B)\mathrm{\ne }\sin A+\sin B$).

• Ignoring domain restrictions in inverse functions.

Practice Problems

1. Solve $2\cos x-1=0$.
   Solution: $x=\pm \frac{\pi }{3}+2\pi n$.

2. Verify ${\cos }^2\theta -{\sin }^2\theta =\cos 2\theta$.

Frequently Asked Questions

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Conclusion

Trigonometry bridges theoretical math and practical problem-solving. Mastering its concepts—from basic ratios to complex identities—empowers professionals in STEM fields and beyond.